eBook

Positive Definite Matrices

Bhatia, Rajendra

2009,2007

Overview

This book represents the first synthesis of the considerable body of new research into positive definite matrices. These matrices play the same role in noncommutative analysis as positive real numbers do in classical analysis. They have theoretical and computational uses across a broad spectrum of disciplines, including calculus, electrical engineering, statistics, physics, numerical analysis, quantum information theory, and geometry. Through detailed explanations and an authoritative and inspiring writing style, Rajendra Bhatia carefully develops general techniques that have wide applications in the study of such matrices. Bhatia introduces several key topics in functional analysis, operator theory, harmonic analysis, and differential geometry--all built around the central theme of positive definite matrices. He discusses positive and completely positive linear maps, and presents major theorems with simple and direct proofs. He examines matrix means and their applications, and shows how to use positive definite functions to derive operator inequalities that he and others proved in recent years. He guides the reader through the differential geometry of the manifold of positive definite matrices, and explains recent work on the geometric mean of several matrices. Positive Definite Matrices is an informative and useful reference book for mathematicians and other researchers and practitioners. The numerous exercises and notes at the end of each chapter also make it the ideal textbook for graduate-level courses.

• Publisher: Princeton University Press
• Subject: Addition / Binomial theorem / Block matrix / Calculation / Cauchy matrix / Cauchy–Schwarz inequality / Coefficient / Commutative property / Completely positive map / Complex number / Convex function / Convex set / Density matrix / Diagonal matrix / Differential geometry / Eigenvalues and eigenvectors / Equation / Equivalence relation / Existential quantification / Extreme point / Fourier transform / Gamma function / General Topics for Engineers / Geometric mean / Geometry / Hadamard product (matrices) / Harmonic analysis / Hermitian matrix / Hilbert space / Hyperbolic function / Infinite divisibility (probability) / Invertible matrix / Lecture / Linear algebra / Linear map / Logarithmic mean / MATHEMATICS / MATHEMATICS / Applied / MATHEMATICS / General / MATHEMATICS / Geometry / General / MATHEMATICS / Mathematical Analysis / Matrices / Matrix (mathematics) / Matrix analysis / Matrix unit / Metric space / Monotonic function / Natural number / Operator algebra / Operator system / Orthonormal basis / Positive element / Positive map / Positive semidefinite / Positive-definite function / Positive-definite matrix / Probability / Probability measure / Projection (linear algebra) / Quantity / Quantum information / Quantum statistical mechanics / Real number / Schur complement / Scientific notation / Self-adjoint operator / Sign (mathematics) / Special case / Spectral theorem / Square root / Standard basis / Summation / Theorem / Unit vector / Unitary matrix / Unitary operator
• ISBN: 9780691129181, 0691129185, 0691029185, 0691168253, 9780691168258, 9781400827787, 1400827787